Multiple lattice tiles and Riesz bases of exponentials

By Mihail N. Kolountzakis. 2013

65 citations recorded by the source corpus.

Suppose $\Omega\subseteq\RR^d$ is a bounded and measurable set and $\Lambda \subseteq \RR^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the $\Lambda$-translates of $\Omega$ cover almost every point of $\RR^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp(2\pi i t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\RR^d$, which forms a Riesz basis of $L^2(\Omega)$. This result was recently proved by Grepstad and Lev under the extra assumption that $\Omega$ has boundary of measure 0, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies $T$ turns out to be a finite union of shifted copies of the dual lattice $\Lambda^*$. It can be chosen knowing only $\Lambda$ and $k$ and is the same for all $\Omega$ that tile multiply with $\Lambda$.

Subfields: Applied Mathematics, Materials Chemistry, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/f680928e3b4efa2c07b6ab8fa52e70b11b98e92c

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In Applied Mathematics

  1. Sigrid Grepstad
  2. Nir Lev

In Materials Chemistry

  1. Sigrid Grepstad
  2. Nir Lev
  3. Mihail N. Kolountzakis

In Numerical Analysis

  1. Sigrid Grepstad
  2. Nir Lev
  3. Mihail N. Kolountzakis

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